I am confused between total derivatives and total differential. What is the difference between total derivatives and total differential?

  • $\begingroup$ For a question like this, you should provide a definition of each of these terms to get a good answer. $\endgroup$ – Bernard W Jul 6 '17 at 1:04

the total differential is

$dz = \frac {\partial z}{\partial x} dx + \frac {\partial z}{\partial y} dy$

How much do we expect $z$ to change for some changes in $x$ and $y?$

The total derivative is

$\frac {dz}{dt} = \frac {\partial z}{\partial x} \frac {dx}{dt} + \frac {\partial z}{\partial y} \frac {dy}{dt}$

$x,y$ are both functions with respect to parameter $t$ what is the derivative of $z$ with respect to this parameter?

  • 1
    $\begingroup$ Notice that in the equation for the total derivative, two different functions are both being called $z$. This is a common abuse of notation, but I think it sometimes causes confusion. $\endgroup$ – littleO Jul 6 '17 at 2:49

Let $f: U \subset \Bbb R^n \to \Bbb R^m$ be differentiable.

The total derivative of $f$ at $a$ is the linear map $df_a$ such that $f(a+t) - f(a) = df_a(t) + o(t)$.

For $m=1$, the total differential of $f$ is

$$df = \sum_{i=1}^m \frac{\partial f}{\partial x_i} dx_i$$

Hope this helps.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.